# Some Recent Algorithms for Finding the Nucleolus of Structured Cooperative Games

## Abstract

Nucleolus is one of the fundamental solution concepts in cooperative game theory. There has been considerable progress in locating the nucleolus in the last three years. The paper motivates through examples how the recent algorithms work efficiently for certain structured class of coperative games. Though the data of a cooperative game grows exponentially in size with the number of players, assignment games, and balanced connected games, grow only polynomially in size, on the number of players. The algorithm for assignment games is based on an efficient graph theoretic algorithm which counts the longest paths to each vertex and trimming of cycles to quickly arrive at the lexicographic geometric centre. Connected games are solved by the technique of feasible direction, initiated in the assignment case. The sellers market corner of the core for assignment games has its counterpart, the lexmin vertex in balanced connected games. Nucleolus has also been characterized via a set of anxioms based on subgame consistency. This is exploited for standard tree games to arrive at an efficient and intuitively explainable algorithm. Improvements on the pivoting manipulations to locate coalitions with constant excess are possible and the paper initially discusses such an algorithm at the beginning.

## Keywords

Cooperative Game Grand Coalition Tree Game Cooperative Game Theory Edge Cost## Preview

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## References

**Bird, C.**[1976] On cost allocation for a spanning tree: a game theoretic approach.Networks: 6, 335–350.Google Scholar**Curiel, I., Pederzoli, G. and S. Tijs.**[1988] Sequencing Games.*European Journal of Operational Research*, 40: 344–351.CrossRefGoogle Scholar**Curiel, I., Potters, J., Rajendra Prasad, V., Tijs, S. and B. Veltman**, [1994] Sequencing and cooperation.*Operations Research*, 42: 366–368.CrossRefGoogle Scholar**Derks, J and J. Kuipers.**[1992] On the core and nucleolus of routing games. Tech Rept. University of Limburg, Maastrict.Google Scholar**Dragan, I.**[1981] A procedure for finding the nucleolus of a cooperative n person game.*Zeitschrift füir Operations Research*, 25: 119–131.Google Scholar**Driessen, T.**[1991] A survey of consistency properties in cooperative game theory.*SIAM Review*, 33: 43–59.CrossRefGoogle Scholar**Galil, Z.**[1980] Application of efficient mergeable heaps for optimization problem on treesActa*Informatica*,13:53–58.CrossRefGoogle Scholar**Granot, D. and F. Granot.**[1992] On some network flow games. Mathematics of Operations Research, 17: 792–841.CrossRefGoogle Scholar**Granot, D. and G. Huberman**[1984] On the core and nucleolus of minimum cost spanning tree games*On the core and nucleolus of minimum cost spanning tree games, mathematical Programming*, 29:323–347.Google Scholar- On some spanning network games. Working paper, The University of British Columbia, Vancouver, British Columbia, Canada.Google Scholar
**Granot, D., Maschler, M., Owen, G. and W. Zhu.**[1996] The kernel/nucleolus of a standard tree game.*International J. Game Theory*, 25:219–244.CrossRefGoogle Scholar**Huberman, G.**[1980] The nucleolus and the essential coalitions.*Analysis and Optimization of Systems*, Springer, Berlin. 416–422.Google Scholar**Kohlberg, E.**[1972] The nucleolus as solution of a minimization problem.*SIAM Journal of Applied Mathematics*, 23: 34–39.CrossRefGoogle Scholar**Kuhn, H.**[1955] The Hungarian Method for assignment problem,Naval*Reserach Logistic Quarterly*, 2:83–97.CrossRefGoogle Scholar**Kuipers, J.**[1994]*Combinatorial methods in cooperative game theory*. Ph.D. Thesis, Maastricht.Google Scholar**Littlechild, S.C. and G. Owen.**[1977] A further note on the nucleolus of the airport game*International J. Game Theory*, 5:91–95.CrossRefGoogle Scholar**Maschler, M.**[1992] The bargaining set, kernel, and nucleolus.In: Aumann, R.J. and S. Hart*(eds).Handbook of Game Theory*. Vol. I.Elsevier science Publ.. BV Amsterdam, North Holland. 591–667.Google Scholar**Maschler, M., Peleg, B. and Shapley, L.**[1972] The kernel and the bargaining set for convex games.*International J. Game Theory*, 1: 73–93.CrossRefGoogle Scholar**Maschler, M., Peleg, B. and Shapley, L.**[1979] Geometric properties of the kernel, nucleolus, and related solution concepts.*Mathematics of Operations Research*4: 303–338.CrossRefGoogle Scholar**Megiddo, N.**[1978] Computational complexity of the game theory approach to cost allocation for a tree*Mathematics of Operations Research*, 3:189–196.CrossRefGoogle Scholar**Noltmier, H.**[1975] An algorithm for the determination of the longest distances in*a graph.Mathematical Programming*, 9: 350–357.CrossRefGoogle Scholar**Owen, G.**[1974] A note on the nucleolus.*International Journal of Game Theory*, 3: 101–103.CrossRefGoogle Scholar**Potters, J., Reijnierse, J. and Ansing, M.**[1996] Computing the nucleolus by solving a prolonged simplex algorithm.*Mathematics of Operations Research*, 21:757–768.CrossRefGoogle Scholar**Sankaran, J.**[1991] On finding the nucleolus of an n-person cooperative game.*International Journal of Game Theory*, 19: 329–338.CrossRefGoogle Scholar**Schmeidler, D.**[1969] The nucleolus of a characteristic function game.*SIAM Journal of Applied Mathematics*, 17: 1163–1170.CrossRefGoogle Scholar**Shapley, L.**[1953] A value for n-person games.*Contributions to the theory of games II*, (Eds. H. Kuhn and A.W. Tucker). Princeton University Press, Princeton, New Jersey, 307–317.Google Scholar**Shapley, L. and Shubik, M.**[1972] The assignment game I: the core.*International Journal of Game Theory*, 1: 111–130.CrossRefGoogle Scholar**Sobolev, A.**[1975] A characterization of optimality principles in cooperative games by functional equations (Russian).*Mathematical Methods in the Social Sciences*, 6: 94–151.Google Scholar**Solymosi, T.**[1993]*On computing the nucleolus of cooperative games*. Ph.D. Thesis, University of Illinois at Chicago.Google Scholar**Solymosi, T. and Raghavan, T.**[1994] An algorithm for finding the nucleolus of assignment games.*International Journal of Game Theory*, 23: 119–143.CrossRefGoogle Scholar**Solymosi, T, Aarts, H and T. Driessen.**[1994] On computing the nucleolus of a balanced connected game. Tech Rept. University of Twente.Google Scholar